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A387322 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

2, 4, 7, 1, 2, 9, 0, 5, 4, 5, 6, 4, 6, 9, 7, 8, 5, 7, 5, 4, 7, 3, 2, 5, 4, 7, 9, 6, 1, 5, 5, 2, 5, 3, 7, 9, 9, 4, 8, 5, 7, 4, 9, 3, 3, 3, 0, 8, 8, 6, 0, 0, 4, 9, 0, 5, 5, 9, 0, 9, 1, 7, 6, 3, 3, 7, 9, 5, 6, 7, 4, 2, 7, 0, 4, 6, 5, 3, 8, 4, 9, 4, 3, 2, 1, 6, 9, 2, 5, 4

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and a square face at the edge where the antiprism and cupola parts of the solid meet.

Also the analogous dihedral angle in a gyroelongated square bicupola (Johnson solid J_45).

			2.4712905456469785754732547961552537994857493330886...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square bicupola.
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square bicupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387323.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A385258 (J_45 volume), A385259 (J_45 surface area).
Cf. A003881, A002193, A010487.

First[RealDigits[Pi/4 + ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[RankedMax[Union[PolyhedronData["J23", "DihedralAngles"]], 4], 10, 100]]

Equals Pi/4 + arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = A003881 + arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A003881 + A387323.

A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

Original entry on oeis.org

1, 6, 8, 5, 8, 9, 2, 3, 8, 2, 2, 4, 9, 5, 3, 0, 2, 6, 5, 8, 5, 7, 5, 9, 3, 9, 5, 0, 3, 3, 5, 3, 7, 8, 0, 7, 8, 4, 3, 6, 4, 5, 6, 9, 8, 3, 2, 4, 4, 8, 2, 4, 0, 3, 5, 3, 1, 5, 3, 5, 5, 6, 1, 5, 3, 0, 2, 6, 1, 3, 3, 2, 5, 4, 7, 4, 9, 8, 6, 2, 4, 4, 6, 6, 4, 6, 8, 3, 8, 3

Offset: 1

Paolo Xausa, Aug 29 2025

This is the dihedral angle between a triangular face and the octagonal face.

			1.6858923822495302658575939503353780784364569832448...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Polytope Wiki, Gyroelongated square cupola.
Wikipedia, Gyroelongated square cupola.

Cf. other J_23 dihedral angles: A177870, A195702, A387320, A387321, A387322.
Cf. A384214 (J_23 volume), A384215 (J_23 surface area).
Cf. A002193, A003881, A010487, A195696.

First[RealDigits[ArcCos[-Sqrt[(7 + Sqrt[32] - 2*Sqrt[20 + 14*Sqrt[2]])/3]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["J23", "DihedralAngles"]], 10, 100]]

Equals arccos(-sqrt((7 + 4*sqrt(2) - 2*sqrt(20 + 14*sqrt(2)))/3)) = arccos(-sqrt((7 + A010487 - 2*sqrt(20 + 14*A002193))/3)).

Equals A387321 - A195696 = A387322 - A003881.

A387349 Positions of 0's in A387348.

Original entry on oeis.org

1, 4, 6, 7, 9, 11, 12, 14, 17, 19, 20, 22, 25, 27, 30, 32, 33, 35, 38, 40, 41, 43, 46, 48, 51, 53, 54, 56, 59, 61, 62, 64, 66, 67, 69, 72, 74, 75, 77, 80, 82, 85, 87, 88, 90, 93, 95, 96, 98, 100, 101, 103, 106, 108, 109, 111, 114, 116, 117, 119, 121, 122

Offset: 1

Clark Kimberling, Aug 27 2025

This sequence together with A387350 and A387351 partition the positive integers. Conjecture: the difference sequence (3, 2, 1, 2, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, ... ) has exactly 3 distinct terms.

Cf. A387348, A387350, A387351.

z = 300;
A[n_, k_] := Module[{t, a, b}, t = (1 + Sqrt[5])/2;
a = Floor[n*(t + 1) + 1 + t/2]; b = Round[a*t]; ({b, a} . MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]];
ts = Table[A[n, k], {n, 0, z - 1}, {k, 0, z - 1}];  (* A035506, Stolarsky array *)
W[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
tw = Table[W[n, k], {n, 1, z}, {k, 1, z}];   (* A035513, Wythoff array *)
diff = tw - ts;
u = Table[diff[[n]][[2]], {n, 1, z}]
Flatten[Position[u, 0]]   (* A387349 *)
Flatten[Position[u, 1]]   (* A387350 *)
Flatten[Position[u, -1]]  (* A387351 *)

A387350 Positions of 1's in A387348.

Original entry on oeis.org

2, 5, 10, 13, 15, 18, 23, 26, 28, 31, 34, 36, 39, 44, 47, 49, 52, 57, 60, 65, 68, 70, 73, 78, 81, 83, 86, 89, 91, 94, 99, 102, 104, 107, 112, 115, 120, 123, 125, 128, 133, 136, 138, 141, 146, 149, 154, 157, 159, 162, 167, 170, 172, 175, 178, 180, 183, 188

Offset: 1

Clark Kimberling, Aug 27 2025

This sequence together with A387349 and A387351 partition the positive integers.

Conjecture: the difference sequence, (3, 5, 3, 2, 3, 5, 3, 2, 3, 3, 2, 3, 5, 3, 2, 3, 5, 3, 5, ... ) has exactly 3 distinct terms.

Cf. A387348, A387349, A387351.

z = 300;
A[n_, k_] := Module[{t, a, b}, t = (1 + Sqrt[5])/2;
a = Floor[n*(t + 1) + 1 + t/2]; b = Round[a*t]; ({b, a} . MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]];
ts = Table[A[n, k], {n, 0, z - 1}, {k, 0, z - 1}];  (* A035506, Stolarsky array *)
W[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
tw = Table[W[n, k], {n, 1, z}, {k, 1, z}];   (* A035513, Wythoff array *)
diff = tw - ts;
u = Table[diff[[n]][[2]], {n, 1, z}]
Flatten[Position[u, 0]]   (* A387349 *)
Flatten[Position[u, 1]]   (* A387350 *)
Flatten[Position[u, -1]]  (* A387351 *)

A387351 Positions of -1's in A387348.

Original entry on oeis.org

3, 8, 16, 21, 24, 29, 37, 42, 45, 50, 55, 58, 63, 71, 76, 79, 84, 92, 97, 105, 110, 113, 118, 126, 131, 134, 139, 144, 147, 152, 160, 165, 168, 173, 181, 186, 194, 199, 202, 207, 215, 220, 223, 228, 236, 241, 249, 254, 257, 262, 270, 275, 278, 283, 288, 291

Offset: 1

Clark Kimberling, Aug 27 2025

This sequence together with A387349 and A387350 partition the positive integers.

Conjecture: the difference sequence, (5, 8, 5, 3, 5, 8, 5, 3, 5, 5, 3, 5, 8, 5, 3, 5, 8, 5, 8, ... ) has exactly 3 distinct terms.

Cf. A387348, A387349, A387350.

z = 300;
A[n_, k_] := Module[{t, a, b}, t = (1 + Sqrt[5])/2;
a = Floor[n*(t + 1) + 1 + t/2]; b = Round[a*t]; ({b, a} . MatrixPower[{{1, 1}, {1, 0}}, k])[[2]]];
ts = Table[A[n, k], {n, 0, z - 1}, {k, 0, z - 1}];  (* A035506, Stolarsky array *)
W[n_, k_] := Fibonacci[k + 1]  Floor[n*GoldenRatio] + (n - 1)  Fibonacci[k];
tw = Table[W[n, k], {n, 1, z}, {k, 1, z}];   (* A035513, Wythoff array *)
diff = tw - ts;
u = Table[diff[[n]][[2]], {n, 1, z}]
Flatten[Position[u, 0]]   (* A387349 *)
Flatten[Position[u, 1]]   (* A387350 *)
Flatten[Position[u, -1]]  (* A387351 *)

A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 639, 1023, 2047, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 126975, 131071, 204799, 229375, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A004767, A048250.

For similar sequences, see A336700, A387411, A387415, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
isA387410(n) = !((1+A000265(A048250(n)))%A000265(1+n));

A387412 The length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 3, 5, 2, 1, 1, 2, 1, 1, 4, 3, 2, 2, 3, 2, 1, 1, 5, 1, 3, 1, 2, 3, 2, 3, 4, 4, 3, 1, 3, 3, 1, 4, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 3, 1, 1, 2, 1, 2, 1, 1, 4, 2, 3, 1, 3, 2, 4, 1, 1, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 4, 2, 6, 2, 3, 1, 4, 1, 1, 3, 5, 1, 3, 2, 3

Offset: 1

Antti Karttunen, Sep 01 2025

Cf. A000523, A003961, A387413.

Cf. also A387422.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A387412(n) = { my(a=binary(n), b=binary(A003961(n)), i=1); while(i<=#a,if(a[i]!=b[i],return(i-1)); i++); (#a); };

from os.path import commonprefix
from math import prod
from sympy import factorint, nextprime
def A387412(n): return len(commonprefix([bin(n)[2:],bin(prod(nextprime(p)**e for p, e in factorint(n).items()))[2:]])) # Chai Wah Wu, Sep 03 2025

a(n) = (1+A000523(n)) - A387413(n).

A387415 Numbers k such that the odd part of (1+k) divides (1 + odd part of A001615(k)), where A001615 is Dedekind's psi-function.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 2431, 4095, 8191, 14335, 16383, 27135, 32767, 44031, 57855, 65535, 75775, 131071, 204799, 262143, 376831, 524287, 667135, 923647, 1048575, 1441791, 1632255, 2056191, 2097151, 2315775, 2744319, 4194303, 6768639, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A001615.

For similar sequences, see A336700, A387410, A387418, A387419.

A000265(n) = (n>>valuation(n,2));
A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
isA387415(n) = !((1+A000265(A001615(n)))%A000265(1+n));

A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 1791, 2047, 2431, 4095, 8191, 14335, 14847, 16383, 27391, 32767, 44031, 57855, 65535, 114687, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2744319, 4194303, 6815743, 8388607, 8781823, 8978431, 12058623, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. The initial 1 is probably the only term that is not in A004767.

Cf. A000225 (subsequence), A000265, A002827, A004767, A034448.

For similar sequences, see A336700, A387410, A387415, A387419.

A000265(n) = (n>>valuation(n,2));
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
isA387418(n) = !((1+A000265(A034448(n)))%A000265(1+n));

A387419 Numbers k such that the odd part of (1+k) divides (1 + odd part of A003959(k)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 3, 4, 7, 15, 31, 40, 63, 127, 255, 511, 639, 1023, 2047, 2175, 2431, 4095, 5247, 8191, 14335, 16383, 32767, 40959, 44031, 57855, 65535, 90111, 131071, 204799, 262143, 376831, 524287, 923647, 1048575, 1632255, 2056191, 2097151, 2621439, 2744319, 4194303, 6815743, 8388607, 8781823, 16777215, 19922943, 24068095

Offset: 1

Antti Karttunen, Sep 01 2025

Like in many sequences of this type, the criterion seems to strongly select for numbers with a long tail of trailing 1-bits. Terms 1, 4 and 40 are probably the only terms that are not in A004767.

Cf. A000225 (subsequence), A000265, A003959, A004767.

For similar sequences, see A336700, A387410, A387411, A387415, A387418.

A000265(n) = (n>>valuation(n,2));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
isA387419(n) = !((1+A000265(A003959(n)))%A000265(1+n));

cp's OEIS Frontend

A387322 Decimal expansion of the fourth largest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387323 Decimal expansion of the smallest dihedral angle, in radians, in a gyroelongated square cupola (Johnson solid J_23).

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A387349 Positions of 0's in A387348.

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A387350 Positions of 1's in A387348.

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A387351 Positions of -1's in A387348.

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A387410 Numbers k such that the odd part of (1+k) divides (1 + odd part of A048250(k)), where A048250 is sum of the squarefree divisors of n.

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A387412 The length of the maximal common prefix of the binary expansions of n and A003961(n), where A003961 is fully multiplicative with a(p) = nextprime(p).

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A387415 Numbers k such that the odd part of (1+k) divides (1 + odd part of A001615(k)), where A001615 is Dedekind's psi-function.

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A387418 Numbers k such that the odd part of (1+k) divides (1 + odd part of A034448(k)), where A034448 is unitary sigma (usigma).

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